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x cos z - xe cos y, y cos z, e² cos y) be the velocity field of a fluid. Compute the flux of v across the surface x² + y² + z² = 4 where x > 0 and the surface is oriented away from the origin. 2 Let v = (4

x cos z - xe cos y, y cos z, e² cos y) be the velocity field of a fluid. Compute the flux of v across the surface x² + y² + z² = 4 where x > 0 and the surface is oriented away from the origin. 2 Let v = (4

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let v = (4 xe cos y, y cos z, e² cos y) be the velocity field of a fluid. Compute the flux of v across the surface x² + y² + z² surface is oriented away from the origin. [₂₁ S X COS Z HINT: Call the surface in this problem S₁. S₁ is "open" and does not enclose a 3D region, so Divergence Theorem cannot be used directly to calculate the flux across S₁. - Instead, try "capping" the S₁ with a disk S₂. Then the surface formed by combining S₁ and S₂ is a "closed" surfce S which does enclose a 3D region. Use the fact that F.dS= = 4 where x > 0 and the S₂ = [F-ds + [₁ S₂ F.dS and calculate FdS by instead calculating S₁ F. dS (using Divergence Theorem) and calculating S FdS (using the original formula).
Transcribed Image Text:Let v = (4 xe cos y, y cos z, e² cos y) be the velocity field of a fluid. Compute the flux of v across the surface x² + y² + z² surface is oriented away from the origin. [₂₁ S X COS Z HINT: Call the surface in this problem S₁. S₁ is "open" and does not enclose a 3D region, so Divergence Theorem cannot be used directly to calculate the flux across S₁. - Instead, try "capping" the S₁ with a disk S₂. Then the surface formed by combining S₁ and S₂ is a "closed" surfce S which does enclose a 3D region. Use the fact that F.dS= = 4 where x > 0 and the S₂ = [F-ds + [₁ S₂ F.dS and calculate FdS by instead calculating S₁ F. dS (using Divergence Theorem) and calculating S FdS (using the original formula).
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